Graph theory may be used to characterize the structure of large complex networks, leading to a better understanding of dynamic interactions that exist between the components of the networks. For example, in biological tissue, nodes with similar characteristics tend to cluster together. The pattern of this clustering provides information about the shared properties of the nodes. Information about the function of the nodes may also be derived from the pattern of the clustering. Networks in biological settings may not be random. Pathological cells may tend to self-organize in clusters and exhibit architectural organization. In particular, many types of cancerous cells may self-organize into clusters and exhibit architectural organization. Networks of cancerous cells may be governed by quantifiable organizing principles. Properties of these organizing principles may be visible in, and extractable from, graphs of such networks.
Spatial graphs and tessellations of pathological tissue, including Voronoi (VT), Delaunay (DT), and minimum spanning trees (MST) built using nuclei as vertices may be predictive of disease severity. For example, VT, DT, and MST graphs have been mined for quantitative features that have been useful in grading prostate and breast cancer. However, these conventional topological methods of grading disease severity focus only on local-edge connectivity. Moreover, conventional graphing methods inherently extract only global features. By extracting only global features, conventional methods are not spatially aware and thus fail to exploit information involving local spatial interactions. Furthermore, conventional methods do not make distinctions between nuclear vertices lying in either the stroma or epithelium. Thus, conventional graphs often traverse the stromal and epithelial regions. Conventional graphing methods may therefore provide sub-optimal results when grading disease severity and patient prognosis.